LMFDB - Number field 18.18.44452341393601500274658203125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19)

gp: K = bnfinit(y^18 - 6*y^17 - 15*y^16 + 147*y^15 - 55*y^14 - 1146*y^13 + 1591*y^12 + 3345*y^11 - 7872*y^10 - 1610*y^9 + 14585*y^8 - 7315*y^7 - 9200*y^6 + 9975*y^5 - 780*y^4 - 2868*y^3 + 1488*y^2 - 285*y + 19, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19)

$ x^{18} - 6 x^{17} - 15 x^{16} + 147 x^{15} - 55 x^{14} - 1146 x^{13} + 1591 x^{12} + 3345 x^{11} + \cdots + 19 $ x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[18, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$44452341393601500274658203125$ 44452341393601500274658203125 $\medspace = 3^{6}\cdot 5^{15}\cdot 7^{6}\cdot 19^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$39.04$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{1/2}5^{5/6}7^{1/2}19^{1/2}\approx 76.37679875285863$
Ramified primes:	$3$, $5$, $7$, $19$ 3, 5, 7, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{8}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a$, $\frac{1}{5}a^{16}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{246331102025}a^{17}+\frac{20174664678}{246331102025}a^{16}-\frac{4404079253}{246331102025}a^{15}-\frac{2169613908}{49266220405}a^{14}-\frac{2515037024}{49266220405}a^{13}+\frac{18566311664}{246331102025}a^{12}+\frac{84454721247}{246331102025}a^{11}+\frac{32757227298}{246331102025}a^{10}+\frac{3666917686}{9853244081}a^{9}+\frac{3588891724}{9853244081}a^{8}+\frac{5606126201}{49266220405}a^{7}+\frac{6256386584}{49266220405}a^{6}+\frac{555027439}{9853244081}a^{5}+\frac{24315206288}{49266220405}a^{4}+\frac{14352061189}{49266220405}a^{3}+\frac{84683153792}{246331102025}a^{2}+\frac{119327743576}{246331102025}a+\frac{9524897299}{246331102025}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/5*a^12 - 2/5*a^11 + 2/5*a^10 - 2/5*a^7 - 1/5*a^6 + 1/5*a^5 + 2/5*a^2 + 1/5*a - 1/5, 1/5*a^13 - 2/5*a^11 - 1/5*a^10 - 2/5*a^8 - 1/5*a^6 + 2/5*a^5 + 2/5*a^3 + 1/5*a - 2/5, 1/5*a^14 - 1/5*a^10 - 2/5*a^9 + 2/5*a^5 + 2/5*a^4 - 2/5, 1/5*a^15 - 1/5*a^11 - 2/5*a^10 + 2/5*a^6 + 2/5*a^5 - 2/5*a, 1/5*a^16 + 1/5*a^11 + 2/5*a^10 + 1/5*a^6 + 1/5*a^5 + 1/5*a - 1/5, 1/246331102025*a^17 + 20174664678/246331102025*a^16 - 4404079253/246331102025*a^15 - 2169613908/49266220405*a^14 - 2515037024/49266220405*a^13 + 18566311664/246331102025*a^12 + 84454721247/246331102025*a^11 + 32757227298/246331102025*a^10 + 3666917686/9853244081*a^9 + 3588891724/9853244081*a^8 + 5606126201/49266220405*a^7 + 6256386584/49266220405*a^6 + 555027439/9853244081*a^5 + 24315206288/49266220405*a^4 + 14352061189/49266220405*a^3 + 84683153792/246331102025*a^2 + 119327743576/246331102025*a + 9524897299/246331102025

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$17$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{13877787454}{49266220405}a^{17}-\frac{63221844043}{49266220405}a^{16}-\frac{310544496164}{49266220405}a^{15}+\frac{1646725597588}{49266220405}a^{14}+\frac{1838400151662}{49266220405}a^{13}-\frac{14658636416751}{49266220405}a^{12}+\frac{12660495887}{9853244081}a^{11}+\frac{58560484637162}{49266220405}a^{10}-\frac{30029589069746}{49266220405}a^{9}-\frac{110007003641154}{49266220405}a^{8}+\frac{87020300066899}{49266220405}a^{7}+\frac{92713960244141}{49266220405}a^{6}-\frac{93476337001071}{49266220405}a^{5}-\frac{26748478678294}{49266220405}a^{4}+\frac{38522766471364}{49266220405}a^{3}-\frac{1331408780246}{49266220405}a^{2}-\frac{4483563139042}{49266220405}a+\frac{671518880273}{49266220405}$, $\frac{323355894543}{49266220405}a^{17}-\frac{1780990119286}{49266220405}a^{16}-\frac{1149028111684}{9853244081}a^{15}+\frac{44779813529582}{49266220405}a^{14}+\frac{4686600881404}{49266220405}a^{13}-\frac{74032373524560}{9853244081}a^{12}+\frac{329251188935323}{49266220405}a^{11}+\frac{12\!\cdots\!41}{49266220405}a^{10}-\frac{19\!\cdots\!49}{49266220405}a^{9}-\frac{15\!\cdots\!93}{49266220405}a^{8}+\frac{39\!\cdots\!14}{49266220405}a^{7}-\frac{295894136497929}{49266220405}a^{6}-\frac{31\!\cdots\!32}{49266220405}a^{5}+\frac{15\!\cdots\!94}{49266220405}a^{4}+\frac{594184795598958}{49266220405}a^{3}-\frac{626330581904218}{49266220405}a^{2}+\frac{152001604583752}{49266220405}a-\frac{11535456675513}{49266220405}$, $a-1$, $\frac{950830717612}{246331102025}a^{17}-\frac{5138235128824}{246331102025}a^{16}-\frac{17292220107521}{246331102025}a^{15}+\frac{25862657279031}{49266220405}a^{14}+\frac{4814020197333}{49266220405}a^{13}-\frac{10\!\cdots\!52}{246331102025}a^{12}+\frac{877885149733844}{246331102025}a^{11}+\frac{36\!\cdots\!21}{246331102025}a^{10}-\frac{10\!\cdots\!04}{49266220405}a^{9}-\frac{911986512532812}{49266220405}a^{8}+\frac{442458712503281}{9853244081}a^{7}-\frac{22596662829016}{9853244081}a^{6}-\frac{17\!\cdots\!29}{49266220405}a^{5}+\frac{172449889317502}{9853244081}a^{4}+\frac{64912541944323}{9853244081}a^{3}-\frac{17\!\cdots\!86}{246331102025}a^{2}+\frac{438481409438657}{246331102025}a-\frac{36519563377337}{246331102025}$, $\frac{951230032871}{246331102025}a^{17}-\frac{5115839788387}{246331102025}a^{16}-\frac{17320493114933}{246331102025}a^{15}+\frac{25700902447128}{49266220405}a^{14}+\frac{4933281696326}{49266220405}a^{13}-\frac{10\!\cdots\!91}{246331102025}a^{12}+\frac{871919756490927}{246331102025}a^{11}+\frac{36\!\cdots\!73}{246331102025}a^{10}-\frac{10\!\cdots\!92}{49266220405}a^{9}-\frac{173805574414668}{9853244081}a^{8}+\frac{437673138243463}{9853244081}a^{7}-\frac{179962320032892}{49266220405}a^{6}-\frac{17\!\cdots\!98}{49266220405}a^{5}+\frac{181652211037345}{9853244081}a^{4}+\frac{291623700706314}{49266220405}a^{3}-\frac{17\!\cdots\!63}{246331102025}a^{2}+\frac{482730613763426}{246331102025}a-\frac{41813783031021}{246331102025}$, $\frac{246405999359}{246331102025}a^{17}-\frac{1360111669098}{246331102025}a^{16}-\frac{4329672716872}{246331102025}a^{15}+\frac{6813296331197}{49266220405}a^{14}+\frac{466170488749}{49266220405}a^{13}-\frac{279295565968829}{246331102025}a^{12}+\frac{261469179595878}{246331102025}a^{11}+\frac{932500175765892}{246331102025}a^{10}-\frac{300014532676333}{49266220405}a^{9}-\frac{41972589797039}{9853244081}a^{8}+\frac{615876962403426}{49266220405}a^{7}-\frac{89106348057136}{49266220405}a^{6}-\frac{483570806504841}{49266220405}a^{5}+\frac{55432728430261}{9853244081}a^{4}+\frac{79890309902101}{49266220405}a^{3}-\frac{524659237438457}{246331102025}a^{2}+\frac{138044523404794}{246331102025}a-\frac{11174112550544}{246331102025}$, $\frac{12793864059}{14490064825}a^{17}-\frac{73546710603}{14490064825}a^{16}-\frac{212585907062}{14490064825}a^{15}+\frac{73439330832}{579602593}a^{14}-\frac{38160018804}{2898012965}a^{13}-\frac{14932386536624}{14490064825}a^{12}+\frac{16253537130133}{14490064825}a^{11}+\frac{48795439315297}{14490064825}a^{10}-\frac{17593573943144}{2898012965}a^{9}-\frac{9994598359554}{2898012965}a^{8}+\frac{35351828577089}{2898012965}a^{7}-\frac{7346843682216}{2898012965}a^{6}-\frac{27239726278247}{2898012965}a^{5}+\frac{16997354315776}{2898012965}a^{4}+\frac{828506473412}{579602593}a^{3}-\frac{30859789495072}{14490064825}a^{2}+\frac{8541333244034}{14490064825}a-\frac{715516707219}{14490064825}$, $\frac{421811131497}{49266220405}a^{17}-\frac{457228761147}{9853244081}a^{16}-\frac{7640484178814}{49266220405}a^{15}+\frac{57518979994989}{49266220405}a^{14}+\frac{9869572004807}{49266220405}a^{13}-\frac{476274465825311}{49266220405}a^{12}+\frac{79352849688164}{9853244081}a^{11}+\frac{16\!\cdots\!71}{49266220405}a^{10}-\frac{23\!\cdots\!03}{49266220405}a^{9}-\frac{20\!\cdots\!24}{49266220405}a^{8}+\frac{49\!\cdots\!93}{49266220405}a^{7}-\frac{56323398957205}{9853244081}a^{6}-\frac{39\!\cdots\!14}{49266220405}a^{5}+\frac{19\!\cdots\!38}{49266220405}a^{4}+\frac{740702420221249}{49266220405}a^{3}-\frac{780293868146704}{49266220405}a^{2}+\frac{38464162472610}{9853244081}a-\frac{14866289653819}{49266220405}$, $\frac{148194803918}{246331102025}a^{17}-\frac{725240358351}{246331102025}a^{16}-\frac{2991891831474}{246331102025}a^{15}+\frac{3662794739879}{49266220405}a^{14}+\frac{2284838580843}{49266220405}a^{13}-\frac{153099233969563}{246331102025}a^{12}+\frac{69276416855391}{246331102025}a^{11}+\frac{537633650974324}{246331102025}a^{10}-\frac{111711492237076}{49266220405}a^{9}-\frac{29223472295205}{9853244081}a^{8}+\frac{246885253597704}{49266220405}a^{7}+\frac{18402005482746}{49266220405}a^{6}-\frac{197455475555661}{49266220405}a^{5}+\frac{18468264550292}{9853244081}a^{4}+\frac{31038104297757}{49266220405}a^{3}-\frac{200763950296849}{246331102025}a^{2}+\frac{61251348985488}{246331102025}a-\frac{5389016267103}{246331102025}$, $\frac{116297932917}{49266220405}a^{17}-\frac{604838066728}{49266220405}a^{16}-\frac{2217703903902}{49266220405}a^{15}+\frac{15280643052813}{49266220405}a^{14}+\frac{5582849679506}{49266220405}a^{13}-\frac{127725816451018}{49266220405}a^{12}+\frac{84449793977176}{49266220405}a^{11}+\frac{447161593087214}{49266220405}a^{10}-\frac{559502921051436}{49266220405}a^{9}-\frac{599360421700457}{49266220405}a^{8}+\frac{12\!\cdots\!17}{49266220405}a^{7}+\frac{51982863893234}{49266220405}a^{6}-\frac{981574414504234}{49266220405}a^{5}+\frac{410209327084451}{49266220405}a^{4}+\frac{191916642241412}{49266220405}a^{3}-\frac{177928591922293}{49266220405}a^{2}+\frac{8686631194357}{9853244081}a-\frac{677937180270}{9853244081}$, $\frac{351688613616}{246331102025}a^{17}-\frac{1730604098397}{246331102025}a^{16}-\frac{7235368290133}{246331102025}a^{15}+\frac{8877053250228}{49266220405}a^{14}+\frac{6056525372784}{49266220405}a^{13}-\frac{382932420527586}{246331102025}a^{12}+\frac{142822006661182}{246331102025}a^{11}+\frac{14\!\cdots\!58}{246331102025}a^{10}-\frac{258867243669007}{49266220405}a^{9}-\frac{468621170484851}{49266220405}a^{8}+\frac{123511362439994}{9853244081}a^{7}+\frac{52102510105592}{9853244081}a^{6}-\frac{577393394164162}{49266220405}a^{5}+\frac{10363932155576}{9853244081}a^{4}+\frac{36632488026520}{9853244081}a^{3}-\frac{288926805463923}{246331102025}a^{2}-\frac{23171811319554}{246331102025}a+\frac{8020187851724}{246331102025}$, $\frac{1069604119489}{246331102025}a^{17}-\frac{5772212723288}{246331102025}a^{16}-\frac{19514054819007}{246331102025}a^{15}+\frac{29086108158931}{49266220405}a^{14}+\frac{5704277067361}{49266220405}a^{13}-\frac{12\!\cdots\!64}{246331102025}a^{12}+\frac{977536751377543}{246331102025}a^{11}+\frac{41\!\cdots\!72}{246331102025}a^{10}-\frac{11\!\cdots\!36}{49266220405}a^{9}-\frac{10\!\cdots\!94}{49266220405}a^{8}+\frac{24\!\cdots\!13}{49266220405}a^{7}-\frac{71879127118196}{49266220405}a^{6}-\frac{20\!\cdots\!79}{49266220405}a^{5}+\frac{926681880070438}{49266220405}a^{4}+\frac{81175361725465}{9853244081}a^{3}-\frac{19\!\cdots\!57}{246331102025}a^{2}+\frac{430320590127664}{246331102025}a-\frac{30305522655149}{246331102025}$, $\frac{754186873548}{246331102025}a^{17}-\frac{4346184071671}{246331102025}a^{16}-\frac{12683712119044}{246331102025}a^{15}+\frac{4368648694279}{9853244081}a^{14}-\frac{1496277303366}{49266220405}a^{13}-\frac{900969400111563}{246331102025}a^{12}+\frac{928625688076776}{246331102025}a^{11}+\frac{30\!\cdots\!54}{246331102025}a^{10}-\frac{10\!\cdots\!93}{49266220405}a^{9}-\frac{717276478428012}{49266220405}a^{8}+\frac{20\!\cdots\!87}{49266220405}a^{7}-\frac{40979864689836}{9853244081}a^{6}-\frac{332900660778691}{9853244081}a^{5}+\frac{825078494993802}{49266220405}a^{4}+\frac{320108497494184}{49266220405}a^{3}-\frac{16\!\cdots\!29}{246331102025}a^{2}+\frac{370417956870553}{246331102025}a-\frac{25810345816423}{246331102025}$, $\frac{20044880681}{49266220405}a^{17}-\frac{102377684354}{49266220405}a^{16}-\frac{78661831630}{9853244081}a^{15}+\frac{2601678461632}{49266220405}a^{14}+\frac{1245308005533}{49266220405}a^{13}-\frac{22016257359879}{49266220405}a^{12}+\frac{12139285603532}{49266220405}a^{11}+\frac{79216353768142}{49266220405}a^{10}-\frac{87663765840214}{49266220405}a^{9}-\frac{115387229949691}{49266220405}a^{8}+\frac{194229975470151}{49266220405}a^{7}+\frac{34199307575729}{49266220405}a^{6}-\frac{165179408531944}{49266220405}a^{5}+\frac{49347440685484}{49266220405}a^{4}+\frac{38470085637826}{49266220405}a^{3}-\frac{25133710870594}{49266220405}a^{2}+\frac{4675954525068}{49266220405}a-\frac{312944182031}{49266220405}$, $\frac{1022179141916}{246331102025}a^{17}-\frac{5557357285417}{246331102025}a^{16}-\frac{18425369151803}{246331102025}a^{15}+\frac{27943392243308}{49266220405}a^{14}+\frac{4328142442547}{49266220405}a^{13}-\frac{11\!\cdots\!36}{246331102025}a^{12}+\frac{980597882753902}{246331102025}a^{11}+\frac{39\!\cdots\!93}{246331102025}a^{10}-\frac{11\!\cdots\!62}{49266220405}a^{9}-\frac{957942029417857}{49266220405}a^{8}+\frac{485100682648700}{9853244081}a^{7}-\frac{35115749561814}{9853244081}a^{6}-\frac{19\!\cdots\!53}{49266220405}a^{5}+\frac{195352469923476}{9853244081}a^{4}+\frac{347952007267166}{49266220405}a^{3}-\frac{19\!\cdots\!73}{246331102025}a^{2}+\frac{491111805666731}{246331102025}a-\frac{38583770179386}{246331102025}$, $\frac{33940121042}{246331102025}a^{17}-\frac{15739562364}{246331102025}a^{16}-\frac{1217305749636}{246331102025}a^{15}+\frac{67781802294}{49266220405}a^{14}+\frac{3274523859229}{49266220405}a^{13}-\frac{2473692133347}{246331102025}a^{12}-\frac{105598655684731}{246331102025}a^{11}+\frac{11933009175901}{246331102025}a^{10}+\frac{14279640198725}{9853244081}a^{9}-\frac{8652719415689}{49266220405}a^{8}-\frac{128367496211949}{49266220405}a^{7}+\frac{18288018371816}{49266220405}a^{6}+\frac{23791099976813}{9853244081}a^{5}-\frac{20005424675644}{49266220405}a^{4}-\frac{51683441928463}{49266220405}a^{3}+\frac{51086250579544}{246331102025}a^{2}+\frac{40880072189622}{246331102025}a-\frac{9503028088837}{246331102025}$, $\frac{2110744447524}{246331102025}a^{17}-\frac{11458587639108}{246331102025}a^{16}-\frac{38205963468382}{246331102025}a^{15}+\frac{57688398036968}{49266220405}a^{14}+\frac{1946657844606}{9853244081}a^{13}-\frac{23\!\cdots\!24}{246331102025}a^{12}+\frac{19\!\cdots\!43}{246331102025}a^{11}+\frac{81\!\cdots\!87}{246331102025}a^{10}-\frac{477294575483971}{9853244081}a^{9}-\frac{20\!\cdots\!12}{49266220405}a^{8}+\frac{49\!\cdots\!13}{49266220405}a^{7}-\frac{243325939507988}{49266220405}a^{6}-\frac{803535941590440}{9853244081}a^{5}+\frac{19\!\cdots\!42}{49266220405}a^{4}+\frac{762106686605908}{49266220405}a^{3}-\frac{38\!\cdots\!87}{246331102025}a^{2}+\frac{936708237064684}{246331102025}a-\frac{70818012650144}{246331102025}$ 13877787454/49266220405a^17 - 63221844043/49266220405a^16 - 310544496164/49266220405a^15 + 1646725597588/49266220405a^14 + 1838400151662/49266220405a^13 - 14658636416751/49266220405a^12 + 12660495887/9853244081a^11 + 58560484637162/49266220405a^10 - 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9994598359554/2898012965a^8 + 35351828577089/2898012965a^7 - 7346843682216/2898012965a^6 - 27239726278247/2898012965a^5 + 16997354315776/2898012965a^4 + 828506473412/579602593a^3 - 30859789495072/14490064825a^2 + 8541333244034/14490064825a - 715516707219/14490064825, 421811131497/49266220405a^17 - 457228761147/9853244081a^16 - 7640484178814/49266220405a^15 + 57518979994989/49266220405a^14 + 9869572004807/49266220405a^13 - 476274465825311/49266220405a^12 + 79352849688164/9853244081a^11 + 1629100879297271/49266220405a^10 - 2377590282482503/49266220405a^9 - 2012828674447624/49266220405a^8 + 4966152036537893/49266220405a^7 - 56323398957205/9853244081a^6 - 3985488945341914/49266220405a^5 + 1944958105133838/49266220405a^4 + 740702420221249/49266220405a^3 - 780293868146704/49266220405a^2 + 38464162472610/9853244081a - 14866289653819/49266220405, 148194803918/246331102025a^17 - 725240358351/246331102025a^16 - 2991891831474/246331102025a^15 + 3662794739879/49266220405a^14 + 2284838580843/49266220405a^13 - 153099233969563/246331102025a^12 + 69276416855391/246331102025a^11 + 537633650974324/246331102025a^10 - 111711492237076/49266220405a^9 - 29223472295205/9853244081a^8 + 246885253597704/49266220405a^7 + 18402005482746/49266220405a^6 - 197455475555661/49266220405a^5 + 18468264550292/9853244081a^4 + 31038104297757/49266220405a^3 - 200763950296849/246331102025a^2 + 61251348985488/246331102025a - 5389016267103/246331102025, 116297932917/49266220405a^17 - 604838066728/49266220405a^16 - 2217703903902/49266220405a^15 + 15280643052813/49266220405a^14 + 5582849679506/49266220405a^13 - 127725816451018/49266220405a^12 + 84449793977176/49266220405a^11 + 447161593087214/49266220405a^10 - 559502921051436/49266220405a^9 - 599360421700457/49266220405a^8 + 1201839061744917/49266220405a^7 + 51982863893234/49266220405a^6 - 981574414504234/49266220405a^5 + 410209327084451/49266220405a^4 + 191916642241412/49266220405a^3 - 177928591922293/49266220405a^2 + 8686631194357/9853244081a - 677937180270/9853244081, 351688613616/246331102025a^17 - 1730604098397/246331102025a^16 - 7235368290133/246331102025a^15 + 8877053250228/49266220405a^14 + 6056525372784/49266220405a^13 - 382932420527586/246331102025a^12 + 142822006661182/246331102025a^11 + 1436770914514058/246331102025a^10 - 258867243669007/49266220405a^9 - 468621170484851/49266220405a^8 + 123511362439994/9853244081a^7 + 52102510105592/9853244081a^6 - 577393394164162/49266220405a^5 + 10363932155576/9853244081a^4 + 36632488026520/9853244081a^3 - 288926805463923/246331102025a^2 - 23171811319554/246331102025a + 8020187851724/246331102025, 1069604119489/246331102025a^17 - 5772212723288/246331102025a^16 - 19514054819007/246331102025a^15 + 29086108158931/49266220405a^14 + 5704277067361/49266220405a^13 - 1207864889110564/246331102025a^12 + 977536751377543/246331102025a^11 + 4161241099458972/246331102025a^10 - 1186869132757336/49266220405a^9 - 1054720727745394/49266220405a^8 + 2497970704223313/49266220405a^7 - 71879127118196/49266220405a^6 - 2032293582454179/49266220405a^5 + 926681880070438/49266220405a^4 + 81175361725465/9853244081a^3 - 1906569358294057/246331102025a^2 + 430320590127664/246331102025a - 30305522655149/246331102025, 754186873548/246331102025a^17 - 4346184071671/246331102025a^16 - 12683712119044/246331102025a^15 + 4368648694279/9853244081a^14 - 1496277303366/49266220405a^13 - 900969400111563/246331102025a^12 + 928625688076776/246331102025a^11 + 3045807075039054/246331102025a^10 - 1020807761777993/49266220405a^9 - 717276478428012/49266220405a^8 + 2083399084565187/49266220405a^7 - 40979864689836/9853244081a^6 - 332900660778691/9853244081a^5 + 825078494993802/49266220405a^4 + 320108497494184/49266220405a^3 - 1612851708395429/246331102025a^2 + 370417956870553/246331102025a - 25810345816423/246331102025, 20044880681/49266220405a^17 - 102377684354/49266220405a^16 - 78661831630/9853244081a^15 + 2601678461632/49266220405a^14 + 1245308005533/49266220405a^13 - 22016257359879/49266220405a^12 + 12139285603532/49266220405a^11 + 79216353768142/49266220405a^10 - 87663765840214/49266220405a^9 - 115387229949691/49266220405a^8 + 194229975470151/49266220405a^7 + 34199307575729/49266220405a^6 - 165179408531944/49266220405a^5 + 49347440685484/49266220405a^4 + 38470085637826/49266220405a^3 - 25133710870594/49266220405a^2 + 4675954525068/49266220405a - 312944182031/49266220405, 1022179141916/246331102025a^17 - 5557357285417/246331102025a^16 - 18425369151803/246331102025a^15 + 27943392243308/49266220405a^14 + 4328142442547/49266220405a^13 - 1154976168163036/246331102025a^12 + 980597882753902/246331102025a^11 + 3934704381432993/246331102025a^10 - 1165838533205262/49266220405a^9 - 957942029417857/49266220405a^8 + 485100682648700/9853244081a^7 - 35115749561814/9853244081a^6 - 1934446707680153/49266220405a^5 + 195352469923476/9853244081a^4 + 347952007267166/49266220405a^3 - 1934918469069073/246331102025a^2 + 491111805666731/246331102025a - 38583770179386/246331102025, 33940121042/246331102025a^17 - 15739562364/246331102025a^16 - 1217305749636/246331102025a^15 + 67781802294/49266220405a^14 + 3274523859229/49266220405a^13 - 2473692133347/246331102025a^12 - 105598655684731/246331102025a^11 + 11933009175901/246331102025a^10 + 14279640198725/9853244081a^9 - 8652719415689/49266220405a^8 - 128367496211949/49266220405a^7 + 18288018371816/49266220405a^6 + 23791099976813/9853244081a^5 - 20005424675644/49266220405a^4 - 51683441928463/49266220405a^3 + 51086250579544/246331102025a^2 + 40880072189622/246331102025a - 9503028088837/246331102025, 2110744447524/246331102025a^17 - 11458587639108/246331102025a^16 - 38205963468382/246331102025a^15 + 57688398036968/49266220405a^14 + 1946657844606/9853244081a^13 - 2390822345134124/246331102025a^12 + 1992406300573143/246331102025a^11 + 8195833794014287/246331102025a^10 - 477294575483971/9853244081a^9 - 2040415747656712/49266220405a^8 + 4989523470974013/49266220405a^7 - 243325939507988/49266220405a^6 - 803535941590440/9853244081a^5 + 1921164675489442/49266220405a^4 + 762106686605908/49266220405a^3 - 3881306530225287/246331102025a^2 + 936708237064684/246331102025a - 70818012650144/246331102025 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 367272638.342 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 367272638.342 \cdot 1}{2\cdot\sqrt{44452341393601500274658203125}}\cr\approx \mathstrut & 0.228323798286 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 - 15*x^16 + 147*x^15 - 55*x^14 - 1146*x^13 + 1591*x^12 + 3345*x^11 - 7872*x^10 - 1610*x^9 + 14585*x^8 - 7315*x^7 - 9200*x^6 + 9975*x^5 - 780*x^4 - 2868*x^3 + 1488*x^2 - 285*x + 19);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$S_3\times D_6$ (as 18T29):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 72

The 18 conjugacy class representatives for $S_3\times D_6$

Character table for $S_3\times D_6$

Intermediate fields

$\Q(\sqrt{5}) $, 3.3.3325.1, 3.3.1425.1, 6.6.55278125.1, 6.6.10153125.1, 9.9.18857855953125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 12 siblings:	data not computed
Degree 18 siblings:	data not computed
Degree 24 sibling:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	12.12.4366065099547265625.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{3}$	R	R	R	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$	${\href{/padicField/13.6.0.1}{6} }^{3}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	R	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$	${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.3.0.1}{3} }^{6}$	${\href{/padicField/43.6.0.1}{6} }^{3}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3	3.6.0.1	$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$5$ 5	5.6.5.1	$x^{6} + 5$	$6$	$1$	$5$	$D_{6}$	$[\ ]_{6}^{2}$
$5$ 5	5.12.10.1	$x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$	$6$	$2$	$10$	$D_6$	$[\ ]_{6}^{2}$
$7$ 7	7.6.0.1	$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$7$ 7	7.12.6.1	$x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$19$ 19	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$[\ ]$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.1	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.1	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.4.2.1	$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	19.4.2.1	$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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